Finite Group Databasebeta

← #27155812

Subgroups — #27155812

Order 12416.

1104 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261467C213yesmaximal, complemented
2Q8×Dic194C211yesmaximal, characteristic, complemented
3#20261443C213yesmaximal, complemented
4#20261404C211yesmaximal, characteristic, complemented
5#20261396C211yesmaximal, characteristic, complemented
6#20261389C213yesmaximal, complemented
7#20261385C213yesmaximal, complemented
8#20261358C213yesmaximal, complemented
9#20261354C213yesmaximal, complemented
10#20261280C213yesmaximal, complemented
11#20261265C213yesmaximal, complemented
12#20261218C211yesmaximal, characteristic, complemented
13#20261217C211yesmaximal, characteristic, complemented
14C2×Q8×Dic97C211yesmaximal, characteristic, complemented
15#548×C97C211yesmaximal, characteristic, complemented
16C2×#13095512C2211yescharacteristic
17C2×#13095510C2216yes—
18#15690810C2213yes—
19#15690809C2213yes—
20#15690809C2213yes—
21#15690806C2213yes—
22#15690805C2213yes—
23#15690803C2213yes—
24#15690800C2213yes—
25#15690799C2213yes—
26#15690796C2213yes—
27#15690796C2216yes—
28#15690794C2213yes—
29#15690794C2216yes—
30#15690792C2216yes—
31#15690791C2213yes—
32#15690790C2216yes—
33#15690787C2216yes—
34#15690787C2211yescharacteristic
35#15690786C2213yes—
36#15690784C2213yes—
37#15690784C2213yes—
38#15690782C2213yes—
39#15690781C2213yes—
40Q8×Dic97C2211yescharacteristic
41Q8×Dic97C2211yescharacteristic
42Q8×Dic97C2213yes—
43Q8×Dic97C2211yescharacteristic
44Q8×Dic97C2213yes—
45C4×Dic194C2213yes—
46C4×D388C2213yes—
47D4×Dic97C2213yes—
48D4×Dic97C2213yes—
49#15690772C2213yes—
50C4×C22⋊D97C2213yes—
51C4×C22⋊D97C2213yes—
52#15690770C2216yes—
53#15690769C2213yes—
54#15690767C2213yes—
55C2×C4⋊Dic97C2213yes—
56C2×C4⋊Dic97C2213yes—
57C4⋊C4×D97C2213yes—
58C2×C4×Dic97C2213yes—
59C2×#41×C97C2211yescharacteristic
60C22×Q8×C97C2211yescharacteristic
61#124×C97C2213yes—
62C22⋊Q8×C97C2213yes—
63#121×C97C2213yes—
64C4×Q8×C97C2211yescharacteristic
65C4×D4×C97C2213yes—
66#13095512—24no—
67#13095510—412no—
68C2×Dic194C2311yescharacteristic
69C2×Dic194C2316yes—
70C2×D388C2313yes—
71C2×C22⋊D97C2313yes—
72C2×C22⋊D97C2313yes—
73C2×C22⋊D97C2316yes—
74C22×Dic97C2311yescharacteristic
75C22×Dic97C2313yes—
76C22×Dic97C2316yes—
77C2×C4×D97C2313yes—
78C2×C4×D97C2316yes—
79C4⋊Dic97C2313yes—
80C4⋊Dic97C2313yes—
81C4⋊Dic97C2313yes—
82C4⋊Dic97C2313yes—
83C4⋊Dic97C2316yes—
84C4⋊Dic97C2311yescharacteristic
85C4⋊Dic97C2313yes—
86#13095490C2313yes—
87#13095490C2311yescharacteristic
88#13095490C2313yes—
89#13095490C2311yescharacteristic
90#13095490C2313yes—
91#13095490C2313yes—
92C22⋊Dic97C2313yes—
93C22⋊Dic97C2313yes—
94C22⋊Dic97C2311yescharacteristic
95C22⋊Dic97C2313yes—
96#13095488C2313yes—
97#13095488C2313yes—
98#13095488C2316yes—
99#13095488C2313yes—
100#13095488C2313yes—
101C4×Dic97C2313yes—
102C4×Dic97C2313yes—
103C4×Dic97C2313yes—
104C4×Dic97C2313yes—
105C4×Dic97C2313yes—
106#41×C97—24no—
107C2×Q8×C97—24no—
108C2×Q8×C97C2313yes—
109C2×Q8×C97C2311yescharacteristic
110C2×D4×C97C2313yes—
111C2×D4×C97C2313yes—
112C22×C4×C97C2313yes—
113C22×C4×C97C2313yes—
114C4⋊C4×C97C2313yes—
115C4⋊C4×C97C2313yes—
116C22⋊C4×C97C2316yes—
117C22⋊C4×C97C2311yescharacteristic
118C22⋊C4×C97C2313yes—
119C42×C97C2313yes—
120C23×C97C2413yes—
121C23×C97C2411yescharacteristic
122C22×D97C2413yes—
123Q8×C97—26no—
124Q8×C97#4114yescomplemented
125D4×C97—26no—
126D4×C97—26no—
127C2×C4×C97—26no—
128C2×C4×C97—26no—
129C2×C4×C97C2411yescharacteristic
130C2×C4×C97C2413yes—
131C2×C4×C97C2413yes—
132C2×C4×C97C2411yescharacteristic
133C2×C4×C97C2413yes—
134C22⋊D97—26no—
135C22⋊D97—26no—
136C22⋊D97—46no—
137C2×Dic97—26no—
138C2×Dic97—22no—
139C2×Dic97—46no—
140C2×Dic97C2411yescharacteristic
141C2×Dic97C2411yescharacteristic
142C2×Dic97C2413yes—
143C2×Dic97C2413yes—
144C2×Dic97C2413yes—
145C2×Dic97C2411yescharacteristic
146C2×Dic97C2411yescharacteristic
147D388—26no—
148C4×D97—26no—
149C4×D97—46no—
150Dic194—22no—
151Dic194—46no—
152C22×C97—21no—
153C22×C97—21no—
154C22×C97—23no—
155C22×C97—23no—
156C22×C97—23no—
157C22×C97H(5,𝔽2)12yes—
158C22×C97C2511yescharacteristic
159C2×D97—23no—
160C2×D97—23no—
161C2×D97—23no—
162C4×C97—21no—
163C4×C97—23no—
164C4×C97—23no—
165C4×C97—21no—
166C4×C97C2×#4116yes—
167Dic97—21no—
168Dic97—23no—
169Dic97—23no—
170Dic97—23no—
171Dic97—21no—
172Dic97—21no—
173Dic972−1+412yes—
174C2×C97—22no—
175C2×C97—43no—
176C2×C97C22×#4111yescharacteristic
177C2×C97C2×H(5,𝔽2)11yescharacteristic
178C2×C97C2×2−1+411yescharacteristic
179D97—43no—
180#3522—971nomaximal, Sylow
181C97#352211yescharacteristic, complemented, Sylow
182#559—973no—
183Q82—971no—
184#554—973no—
185#550—971no—
186#548—971no—
187#548D9711yescharacteristic, complemented
188#547—973no—
189#547—973no—
190#542—973no—
191#542—973no—
192#532—973no—
193#529—973no—
194#520—971no—
195#520—971no—
196C2×C4×Q8—971no—
197C2×#41—971no—
198C2×#41—976no—
199C2×#41C2×D9711yescharacteristic
200C22×Q8C2×D9711yescharacteristic
201C4⋊Q8—973no—
202C4⋊Q8—973no—
203C4⋊Q8—973no—
204#127—973no—
205#126—973no—
206#126—973no—
207#125—973no—
208#124—973no—
209#124—973no—
210#124—976no—
211#124C2×D9713yes—
212#123—973no—
213#123—973no—
214#123—976no—
215#123—976no—
216C22⋊Q8—973no—
217C22⋊Q8—976no—
218C22⋊Q8—971no—
219C22⋊Q8—976no—
220C22⋊Q8C2×D9713yes—
221#121—973no—
222#121—973no—
223#121—973no—
224#121—973no—
225#121C2×D9713yes—
226C4×Q8—971no—
227C4×Q8—973no—
228C4×Q8—971no—
229C4×Q8—973no—
230C4×Q8—971no—
231C4×Q8—973no—
232C4×Q8C2×D9711yescharacteristic
233C4×D4—973no—
234C4×D4—973no—
235C4×D4—973no—
236C4×D4—973no—
237C4×D4—973no—
238C4×D4—976no—
239C4×D4—973no—
240C4×D4C2×D9713yes—
241#117—973no—
242#117—973no—
243C2×C4⋊C4—973no—
244C2×C4⋊C4—973no—
245C2×C4⋊C4—973no—
246C2×C42—973no—
247#41—24no—
248#41—38812no—
249#41—1944no—
250C2×Q8—24no—
251C2×Q8—976no—
252C2×Q8—971no—
253C2×Q8C22×D9713yes—
254C2×Q8C22×D9711yescharacteristic
255C2×D4—976no—
256C2×D4—973no—
257C2×D4—973no—
258C2×D4—973no—
259C2×D4C22×D9713yes—
260C2×D4C22×D9713yes—
261C22×C4—976no—
262C22×C4—971no—
263C22×C4—973no—
264C22×C4—976no—
265C22×C4—973no—
266C22×C4C22×D9713yes—
267C22×C4C22×D9713yes—
268C4⋊C4—973no—
269C4⋊C4—971no—
270C4⋊C4—971no—
271C4⋊C4—973no—
272C4⋊C4—973no—
273C4⋊C4—973no—
274C4⋊C4—973no—
275C4⋊C4—973no—
276C4⋊C4—973no—
277C4⋊C4—973no—
278C4⋊C4—976no—
279C4⋊C4—971no—
280C4⋊C4—973no—
281C4⋊C4C22×D9713yes—
282C4⋊C4C22×D9713yes—
283C22⋊C4—973no—
284C22⋊C4—973no—
285C22⋊C4—971no—
286C22⋊C4—973no—
287C22⋊C4—973no—
288C22⋊C4—973no—
289C22⋊C4—973no—
290C22⋊C4—976no—
291C22⋊C4—973no—
292C22⋊C4C22×D9716yes—
293C22⋊C4C22×D9711yescharacteristic
294C22⋊C4C22×D9713yes—
295C42—973no—
296C42—973no—
297C42—973no—
298C42—973no—
299C42—973no—
300C42C22×D9713yes—
301C23—973no—
302C23C23×D9713yes—
303C23C23×D9711yescharacteristic
304Q8—26no—
305Q8—3886no—
306Q8—1942no—
307Q8#1309551014yescomplemented
308D4—26no—
309D4—26no—
310D4—3886no—
311D4—1946no—
312D4—1946no—
313D4—1946no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—971no—
317C2×C4—971no—
318C2×C4—971no—
319C2×C4—973no—
320C2×C4—973no—
321C2×C4—973no—
322C2×C4—971no—
323C2×C4—3886no—
324C2×C4—3886no—
325C2×C4—1942no—
326C2×C4—1946no—
327C2×C4—1946no—
328C2×C4C23×D9713yes—
329C2×C4C23×D9711yescharacteristic
330C2×C4C23×D9713yes—
331C2×C4C23×D9711yescharacteristic
332C2×C4C23×D9713yes—
333C22—21no—
334C22—21no—
335C22—23no—
336C22—23no—
337C22—23no—
338C22—1943no—
339C22—1943no—
340C22—1943no—
341C22#1569085612yes—
342C22C24×D9711yescharacteristic
343C4—21no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—972no—
348C4—1941no—
349C4—1943no—
350C4—1943no—
351C4—1943no—
352C4—1941no—
353C4—1941no—
354C4C2×#1309551016yes—
355C2—22no—
356C2—43no—
357C2—3883no—
358C2C22×#1309551011yescharacteristic
359C2C2×#1569085611yescharacteristic
360C22−1+4×D9711yescharacteristic
subgroupG/H
center, FrattiniC22C24×D97
commutator, socle, cosocleC22×C97C25
Fitting#548×C97C2
hypercenter#548D97
hypocenterC97#3522
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ #548

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #20261467C2semidirect
2. Q8×Dic194C2semidirect
3. #20261443C2semidirect
4. #20261404C2semidirect
5. #20261396C2semidirect
6. #20261389C2semidirect
7. #20261385C2semidirect
8. #20261358C2semidirect
9. #20261354C2semidirect
10. #20261280C2semidirect
11. #20261265C2semidirect
12. #20261218C2semidirect
13. #20261217C2semidirect
14. C2×Q8×Dic97C2semidirect
15. #548×C97C2semidirect
124. Q8×C97#41semidirect
181. C97#3522semidirect
187. #548D97semidirect
307. Q8#13095510semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202614672
Q8×Dic1942
#202614432
#202614042
#202613962
#202613892
#202613852
#202613582
#202613542
#202612802
#202612652
#202612182
#202612172
C2×Q8×Dic972
#548×C972
#352297